In the previous post Sequential spaces, IV, we show that the uncountable product of sequential spaces is not sequential (e.g. the product is not sequential). What is more remarkable is that the product of two sequential spaces needs not be sequential. We present an example of a first countable space and a Frechet space whose product is not a k-space (thus not sequential). For the previous discussion on this blog on sequential spaces and k-spaces, see the links at the end of this post.
Let be the real line and let be the set of all positive integers. Let be the space with the topology inherited from the usual topology on the real line. Let with the positive integers identified as one point (call this point ). We claim that is not a k-space and thus not a sequential space. To this end, we define a non-closed such that is closed in for all compact .
Let where for each , the set is defined by the following:
Clearly is not closed as . In fact in the product space , the point is the only limit point of the set . Another observation is that for each , is not a limit point of . Furthermore, if for each where is an infinite subset of , then is not a limit point of . It follows that no infinite subset of is compact. Consequently, is finite for each compact . Thus is not a k-space. To see that is not sequential directly, observe that is sequentially closed.
Previous posts on sequential spaces and k-spaces